Accurate measurement of the in-plane magnetic anisotropy energy function Ea(θ)Ea(θ) in ultrathin films by magneto-optics

We present a sensitive and precise determination of an arbitrary thin film magnetic anisotropy energy EaEa (or Ea/MSEa/MS if the saturation magnetization MSMS is not known) versus in-plane polar angle θθ by means of initial inverse susceptibility and torque measurements under constant transverse bias field HBHB. In this method coined TBIISTtwo magnetic fields HLHL and HBHB are applied in the plane of the film along directions defined by θθ and θ-(π/2)θ-(π/2), respectively, and mLmL the magnetization component in units of MSMS is measured versus HLHL around mL=0mL=0. This readily provides two quantities χ-1=(∂HL/∂mL)(mL=0)χ-1=(∂HL/∂mL)(mL=0) the initial inverse susceptibility and ΔH=HL(mL=0)ΔH=HL(mL=0) the field offset or torque at mL=0mL=0, shown to be respectively related to the second (1/MS)(∂2Ea/∂θ2)(θ-(π/2))(1/MS)(∂2Ea/∂θ2)(θ-(π/2)) and first (1/MS)(∂Ea/∂θ)(θ-π/2)(1/MS)(∂Ea/∂θ)(θ-π/2) derivatives of EaEa. This, in turn, yields two determinations of Ea(θ)Ea(θ) from χ-1(θ)χ-1(θ) and ΔH(θ)ΔH(θ), respectively. Fourier analysis then easily resolves various contributions of different symmetries to the magnetic anisotropy. The magnetization mL(HL)mL(HL) is measured with a conventional magneto-optical Kerr experiment in longitudinal geometry with respect to HLHL and higher-order (nonlinear in mLmL) contributions as well as polar or other contributions to the Kerr signal are carefully determined and corrected in order to obtain the desirable accuracy or even simply meaningful results. Typical acquisition times including all corrections are about a few minutes per angle θθ. The power of the method is demonstrated on typical examples of epitaxial Fe layers on Si(0 0 1) and Si(1 1 1) substrates. For instance, it is shown that extremely small contributions up to the 12th order in-spin to the cubic magnetocrystalline anisotropy of iron are readily detected.